15. The equation Ux U x + 1 Ux+2 = a (u2+ux + Ux+2) x+1 may be integrated by assuming u1 = √√/a tan v ̧• 16. Shew also that the general integral of the above equation is included in that of the equation u2-u2=0, and hence deduce the former. 17. Shew how to integrate the equation Ux+1Ux+2 + Ux+2 Ux+UzUx+1 = m2. Их and shew that if m be the integral part of √n, converges as x increases to the decimal part of √n. 19. If a, be a fourth proportional to a, b, c, b, a fourth proportional to b, c, a, and c, to c, a, b, and a,, b, c, depend in the same manner on a,, b, c,, find the linear equation of differences on which a, depends and solve it. 20. Solve the equation x (x + 1) ▲3u2+k (1 − x) ▲ux + ku2 = 0. 21. Solve the equation | U+ U2+1) U2+3 ,, 19 = C 22. If vo, v1, V 2 &c. be a series of quantities the successive terms of which are connected by the general relation and if v, v, be any given quantities, find the value of v. [S. P.] 23. If n integers are taken at random and multiplied together in the denary scale, find the chance that the figure in the unit's place will be 2. where a is one of the imaginary (n+1)th roots of unity, the n+1 constants being subject to an equation of condition. 25. Solve the equation Pn+1 = P2+Pn-1P ̧ + Pm¿P1 + &c. + P¿Pn1+ Pn› 3 n-2 4 4n 6 Ux Xx n [Catalan, Liouville, III. 508.] +Ux-1 and deduce therefrom the solution of Ux+1 = Ux + (x3 − x) Ùx-1° [Sylvester, Phil. Mag.] CHAPTER XIII. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS, 1. THE symbolical methods for the solution of differential equations whether in finite terms or in series (Diff. Equations, Chap. XVII.) are equally applicable to the solution of difference-equations. Both classes of equations admit of the same symbolical form, the elementary symbols combining according to the same ultimate laws. And thus the only remaining difference is one of interpretation, and of processes founded upon interpretation. It is that kind of difference which -1 d exists between the symbols () and . dx It has been shewn that if in a linear differential equation we assume x = €, the equation may be reduced to the form ƒ. (d) u +ƒ. (d) e°u + ƒ2 (d) ¿u d €20 U + ... de U being a function of 0. Moreover, the symbols (2). f €mo = ƒ (m) Єmo And hence it has been shewn to be possible, 1st, to express the solution of (1) in series, 2ndly, to effect by general theorems the most important transformations upon which finite integration depends. Now and e are the equivalents of x and x, and it is dx de proposed to develope in this chapter the corresponding theory of difference-equations founded upon the analogous employ ment of the symbols x; therefore A Ax and xE, supposing Ax arbitrary, and A$ (x) = 4 (x + Ax) − † (x), Ep (x) = $ (x + Ax). PROP. 1. If the symbols π and p be defined by the equations the subject of operation in the second theorem being unity. .(4), 1st. Let Ax=r, and first let us consider the interpretation of pux an equation to which we may also give the form m pTMu2 = x (x + y)..... {x + (m − 1) r} ETMu, ... ... ... ... ... · (5). x r) If u = 1, then, since Umr = 1, we have p”1 = x (x + r) ... {x + (m − 1) r}, to which we shall give the form m pTM = x (x + r) ... {x+(m −1) r}, the subject 1 being understood. (x+r)... (x+mr) Ux+m+1 r — X... {x + (m − 1) r} Ux+mr · x = x = x r {x + (m − 1) r} Em XUx+r — (x — mr) u, r 20 Therefore supposing ƒ(#) a function expressible in ascending powers of π, we have ƒ (π) pTMu = pmƒ (π+m) u............. which is the first of the theorems in question. Again, supposing u=1, we have f" ..(6), = p" { ƒ (m) + ƒ′′ (m) = + " "(m) ~ " + &c.} 1. 1.2 |